2 Specification

3 Description

The system of equations is defined as:

fix1,x2,…,xn=0, for ​i=1,2,…,n.

nag_zero_nonlin_eqns (c05nbc) is based upon the MINPACK routine HYBRD1 (see Moré et al. (1980)). It chooses the correction at each step as a convex combination of the Newton and scaled gradient directions. Under reasonable conditions this guarantees global convergence for starting points far from the solution and a fast rate of convergence. The Jacobian is updated by the rank-1 method of Broyden. At the starting point the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970).

On exit: the function values fix (unless userflag is set to a negative value by f).

4:
userflag – Integer *Input/Output

On entry: userflag>0.

On exit: in general, userflag should not be reset by f. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), then userflag should be set to a negative integer. This value will be returned through fail.errnum.

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_INT_ARG_LE

On entry, n=value.
Constraint: n>0.

NE_NO_IMPROVEMENT

The iteration is not making good progress.
This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 8). Otherwise, rerunning nag_zero_nonlin_eqns (c05nbc) from a different starting point may avoid the region of difficulty.

There have been at least 200*(n+1) evaluations of f().
Consider restarting the calculation from the point held in x.

NE_USER_STOP

User requested termination, user flag value =value.

NE_XTOL_TOO_SMALL

No further improvement in the solution is possible. xtol is too small: xtol=value.

7 Accuracy

If x^ is the true solution, nag_zero_nonlin_eqns (c05nbc) tries to ensure that

x-x^≤xtol×x^.

If this condition is satisfied with xtol=10-k, then the larger components of x have k significant decimal digits. There is a danger that the smaller components of x may have large relative errors, but the fast rate of convergence of nag_zero_nonlin_eqns (c05nbc) usually avoids this possibility.

If xtol is less than machine precision and the above test is satisfied with the machine precision in place of xtol, then the function exits with NE_XTOL_TOO_SMALL.

Note: this convergence test is based purely on relative error, and may not indicate convergence if the solution is very close to the origin.

The test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then nag_zero_nonlin_eqns (c05nbc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns (c05nbc) with a tighter tolerance.

8 Further Comments

The time required by nag_zero_nonlin_eqns (c05nbc) to solve a given problem depends on n, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by nag_zero_nonlin_eqns (c05nbc) to process each call of f is about 11.5×n2. Unless f can be evaluated quickly, the timing of nag_zero_nonlin_eqns (c05nbc) will be strongly influenced by the time spent in f.

Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.

9 Example

This example determines the values x1,…,x9 which satisfy the tridiagonal equations: